► In this thesis we find that all imaginary n-quadratic fields with n>3 have classnumber larger than 1 and therefore cannot be Euclidean. We…
(more)

▼ In this thesis we find that all imaginary n-quadratic fields with n>3 have classnumber larger than 1 and therefore cannot be Euclidean. We also examine imaginary triquadratic fields, presenting a complete list of 17 imaginary triquadratic fields with classnumber 1, and classifing many of them according to whether or not they are norm-Euclidean. We find that at least three of these fields are norm-Euclidean, and at least five are not.
Advisors/Committee Members: Katherine Stange, David Grant, Robert Tubbs, Eric Stade, Franck Vernerey.

► The analytic classnumber formula equates the residue of a dedekind zeta function at 1 with various properties of a related number field. These properties…
(more)

▼ The analytic classnumber formula equates the residue
of a dedekind zeta function at 1 with various properties of a
related number field. These properties are defined, their
correlation explained, and then they are related to the formula.
The classnumber, a measure of a number field's failure to be a
principal ideal domain, is defined and analyzed. Finally, the
formula is explicitly stated and proven.
Advisors/Committee Members: Kamienny, Sheldon (Committee Chair), Guralnick, Robert M. (Committee Member), Lanski, Charles (Committee Member).

► The aim of this thesis is to determine if it is possible, using p-adic techniques, to unconditionally evaluate the p-valuation of the classnumber h…
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▼ The aim of this thesis is to determine if it is possible, using p-adic techniques, to unconditionally evaluate the p-valuation of the classnumber h of an algebraic number field K. This is important in many areas of number theory, especially Iwasawa theory.
The class group ClK of an algebraic number field K is the group of fractional ideals of K modulo principal ideals. Its cardinality (the classnumber h) is directly linked to the existence of unique factorisation in K, and hence the class group is of core importance to almost all multiplicative problems concerning number fields. The explicit computation of ClK (and h) is a fundamental task in computational number theory.
Despite its importance, existing algorithms cannot obtain the class group unconditionally in a reasonable amount of time if the field has a large discriminant. Although faster, specialised algorithms (focused only on calculating the p-valuation) are limited in the cases with which they can deal.
We present two algorithms to verify the p-valuation of h for any totally real abelian number field, with no restrictions on p. Both algorithms are based on the p-adic classnumber formula and work by computing p-adic L-functions Lp(s,χ) at the value of s = 1. These algorithms came about from two different ways of computing Lp(1,χ), using either a closed or a convergent series formula.
We prove that our algorithms compare favourably against existing class group algorithms, with superior complexity for number fields of degree 5 or higher. We also demonstrate that our algorithms are faster in practice. Finally, we present some open questions arising from the algorithms.

► We prove several results concering class groups of number fields and function fields. Firstly we compute all the moments of the p-torsion in the first…
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▼ We prove several results concering class groups of number fields and function fields.
Firstly we compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic number fields of degree p, unconditionally for p=3 and underGRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the p=3 case this gives the distribution of the 3-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the 4-torsion in quadratic fields.
Secondly, we compute all the moments of a normalization of the function which counts unramified H8-extensions of quadratic number fields, where H8 is the quaternion group of order 8, and show that the values of this function determine a point mass distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for several other 2-groups G, which we conjecture will give finite moments which determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified H8-extensions. This part of the thesis is joint work with Brandon Alberts.
Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over ℙ1/𝔽p. This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of A4, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group G for the cases when G is S3, S4, A4, D2l and ℤ/lℤ\rtimesℤ/rℤ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.
Advisors/Committee Members: Tsimerman, Jacob, Mathematics.

► The ideal class group is an important object in the study of algebraic number fields, as is the associated classnumber. In cyclotomic fields in…
(more)

▼ The ideal class group is an important object in the
study of algebraic number fields, as is the associated classnumber. In cyclotomic fields in particular, the classnumber
decomposes into two factors. While the second factor is difficult
to determine due to the need for a basis of the group of units, the
first factor is given by a surprisingly simple formula. In terms of
the structure of the ideal class group, Stickelberger's theorem
gives an important result. The theorem provides explicit
annihilators of the ideal class group, from which the Stickelberger
ideal arises. In this work, the analytic classnumber formula is
derived and related to the minus part of the index of the
Stickelberger ideal in the integral group ring over the Galois
group.
Advisors/Committee Members: Sinnott, Warren (Advisor).

…classnumber, h− , and the minus part of the Stickelberger ideal,
I− . Following Iwasawa, it… …deriving the analytic classnumber formula. To begin, let
σ1 , σ2 , . . . , σr1 be the real… …result which will later be used
in the process of establishing the analytic classnumber… …where f and g are the residue class degree
and the number of primes lying above p in k.
Proof… …result will be central to the determination of a formula for the classnumber. ([3…

Given an odd prime p, A technique due to Jean-Fran\c{c}ois Mestre allows one to construct infinitely many quadratic fields for which the ideal class group…
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▼

Given an odd prime p, A technique due to Jean-Fran\c{c}ois Mestre allows one to construct infinitely many quadratic fields for which the ideal class group has p – rank at least 2, using a degree p isogeny between elliptic curves such that the kernel has a rational point. This technique only works for primes p ≤ 7; we attempt to generalize the construction for larger primes. One line of approach uses higher degree isogenies (which have no rational point in the kernel), from which we obtain higher-degree number fields with p – rank at least two. In the process, we collect data on the number fields generated by the points in the kernel of an isogeny, and make a series of conjectures based on the data. We also discuss the possibilities and limitations of replacing the elliptic curves in Mestre's technique with more general abelian varieties.

► Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K).…
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▼ Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K). It is known, due to a tremendous collection of work ranging from the principal results of class field theory to the famous Golod-Shafarevich inequality, that G is finite if the p-rank of the class group of K is 0 or 1, and is infinite if this rank is at least 3. This leaves the rank 2 case as the only remaining unsolved case. In this case, while finiteness is still a mystery, much is still known about G: It is a 2-generated, 2-related pro-p-group equipped with an involution that acts as the inverse modulo commutators, and is of one of three possible Zassenhaus types (defined in the paper). If such a group is finite, we will call it an interesting p-tower group. We further the knowledge on such groups by showing that one particular Zassenhaus type can occur as an interesting p-tower group only if the group has order at least p24 (Proposition 8.1), and by proving a succinct cohomological condition (Proposition 4.7) for a p-tower group to be infinite. More generally, we prove a Golod-Shafarevich equality (Theorem 5.2), refining the famous Golod-Shafarevich inequality, and obtaining as a corollary a strict strengthening of previous Golod-Shafarevich inequalities (Corollary 5.5). Of interest is that this equality applies not only to finite p-groups but also to p-adic analytic pro-p-groups, a class of groups of particular relevance due to their prominent appearance in the Fontaine-Mazur conjecture. This refined version admits as a consequence that the sizes of the first few modular dimension subgroups of an interesting p-tower group G are completely determined by p and its Zassenhaus type, and we compute these sizes. As another application, we prove a new formula (Corollary 5.3) for the Fp-dimensions of the successive quotients of dimension subgroups of free pro-p-groups.
Advisors/Committee Members: McCallum, William G (advisor).

► There are many computationally difficult problems in the study of p-adic fields, among them the classification of field extensions and the decomposition of global ideals.…
(more)

▼ There are many computationally difficult problems in the study of p-adic fields, among them the classification of field extensions and the decomposition of global ideals. The main goal of this work is present efficient algorithms, leveraging the Newton polygons and residual polynomials, to solve many of these problems faster and more efficiently than present methods. Considering additional invariants, we extend Krasner's mass formula, dramatically improve general extension enumeration using the reduced Eisenstein polynomials of Monge, and provide a detailed account of algorithms that compute Okutsu invariants, which have many uses, through the lens of partitioning the set of zeros of polynomials.

NC DOCKS at The University of North Carolina at Greensboro; Sinclair, B. (2015). Algorithms for enumerating invariants and extensions of local fields. (Thesis). NC Docks. Retrieved from http://libres.uncg.edu/ir/uncg/f/Sinclair_uncg_0154D_11665.pdf

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

NC DOCKS at The University of North Carolina at Greensboro; Sinclair, Brian. “Algorithms for enumerating invariants and extensions of local fields.” 2015. Thesis, NC Docks. Accessed June 07, 2020.
http://libres.uncg.edu/ir/uncg/f/Sinclair_uncg_0154D_11665.pdf.

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

NC DOCKS at The University of North Carolina at Greensboro; Sinclair, Brian. “Algorithms for enumerating invariants and extensions of local fields.” 2015. Web. 07 Jun 2020.

Vancouver:

NC DOCKS at The University of North Carolina at Greensboro; Sinclair B. Algorithms for enumerating invariants and extensions of local fields. [Internet] [Thesis]. NC Docks; 2015. [cited 2020 Jun 07].
Available from: http://libres.uncg.edu/ir/uncg/f/Sinclair_uncg_0154D_11665.pdf.

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

NC DOCKS at The University of North Carolina at Greensboro; Sinclair B. Algorithms for enumerating invariants and extensions of local fields. [Thesis]. NC Docks; 2015. Available from: http://libres.uncg.edu/ir/uncg/f/Sinclair_uncg_0154D_11665.pdf

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

► We study the computation of the structure of two finite abelian groups associated with function fields over finite fields: the degree zero divisor class group…
(more)

▼ We study the computation of the structure of two
finite abelian groups associated with function fields over finite
fields: the degree zero divisor class group and the multiplicative
group of the finite field itself. In addition, we present a novel
algorithm to factor polynomials over finite fields using Carlitz
modules. ❧ Let Fq denote the finite field with q elements and let
k = Fq(t) be the rational function field with constant field Fq.
Let F/k be a finite geometric abelian extension where an
Fq‐rational place in k denoted as ∞ splits completely. Stark units
are certain functions in F whose existence is claimed in the
function field analogue of the Brumer‐Stark conjecture. In the
function field setting, the Brumer‐Stark conjecture and hence the
existence of Stark units was proven by P. Deligne. Further, Stark
units are determined by the evaluations at 0 of Artin's L‐functions
associated with the complex characters of Gal(F/k). We prove that
for all primes ℓ not dividing q[F : k], the structure of the ℓ‐part
of the divisor class group of F is determined by Kolyvagin
derivative classes that are constructed out of Euler systems
associated with the Stark units. ❧ Further, given a certain
ℤ[Gal(F/k)] generator of the Stark units, we describe an algorithm
to compute the structure of the ℓ‐part of the divisor class group.
When F/k is a narrow ray class field (or a small index subextension
of a narrow ray class field), such a generator of the Stark units
module can be efficiently computed. ❧ The divisor class group
Cl0F of F is a finite abelian group and fits in the exact
sequence ❧ 0 → RF → Cl0F → pic(OF) → 0 ❧ where RF is the
regulator and pic(OF) is the ideal class group of OF. Our
algorithm to compute the ℓ‐part of the divisor class group is
heavily reliant on the machinery of Euler systems of Stark units
and is efficient if the ℓ‐part of the ideal class group is small.
Empirical and heuristic evidence point to the ideal class group
being of very small order in comparison to the divisor class group.
❧ Other applications of our technique include a fast algorithm for
computing the divisor classnumber of narrow ray class extensions.
❧ We next turn to computing primitive elements in finite fields of
small characteristic. The multiplicative group of a finite field is
cyclic and generators (primitive elements) are abundant. However,
finding one efficiently remains an unsolved problem. We describe a
deterministic algorithm for finding a generating element of the
multiplicative group of the finite field Fpn with pⁿ elements
where p is a prime. In time polynomial in p and n, the algorithm
either outputs an element that is provably a generator or declares
that it has failed in finding one. Under a heuristic assumption,
the algorithm does succeed in finding a generator. The algorithm
relies on a relation generation technique in a recent breakthrough
by Antoine Joux's for discrete logarithm computation in small
characteristic finite fields. ❧ Building upon Joux's algorithm,
Barbulescu, Gaudry, Joux and Thome…
Advisors/Committee Members: Huang, Ming-Deh (Committee Chair), Adleman, Leonard (Committee Member), Kamienny, Sheldon (Committee Member).

► The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group G, one would like to…
(more)

▼ The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group G, one would like to list all Galois extensions L/F such that the Galois group of L/F is G.
In this work we shall solve this problem for all fields F, and for group G of unipotent 4 × 4 matrices over 𝔽2. We also list all 16 U4 (𝔽2)-extensions of ℚ2. The importance of these results is that they answer the inverse Galois problem in some specific cases.
This is joint work with J\'an Min\'a\v{c} and Nguyen Duy T\an.

▼ ããIn this work, D-optimal designs for binary response models in mixture experiments are discussed. This kind of model setting occurs in many chemical experiments. Under the linear logit link, D-optimal designs for binary response models with two explanatory variables on the rst quadrant as the design space have been investigated by Sitter and Torsney (1995) and Haines et al. (2007).
ããWe first provide an approach to obtain an essentially complete class of designs with reduced number of optimal supports, then the D-optimal designs may also be obtained with more insights on the structure appeared there. It is helpful for the search of D-optimal designs on the other design spaces with more restrictions. Later under the design space with constraints due to the mixture restriction, we obtain the D-optimal designs for binary response models under the linear logit link in a mixture design space.
Advisors/Committee Members: Mei-Hui Guo (chair), Fu-Chuen Chang (chair), May-Ru Chen (chair), Mong-Na Lo Huang (committee member), Kerby Shedden (chair).

► We give a new definition of a p-adic L-function for a mixed signature character of a real quadratic field and for a nontrivial ray class…
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▼ We give a new definition of a p-adic L-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a p-adic Stark conjecture for this p-adic L-function. We prove our conjecture in the case when p is split in the imaginary quadratic field by relating our construction to Katz's p-adic L-function. We also prove our conjecture in the real quadratic setting for one special case and give numerical evidence in one specific example.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a…
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▼

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known.
Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work.
In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 classnumber power without computing class numbers.

► In 1997, Xian-Jin Li gave an equivalence to the classical Riemann hypothesis, now referred to as Li's criterion, in terms of the non-negativity of a…
(more)

▼ In 1997, Xian-Jin Li gave an equivalence to the classical Riemann hypothesis,
now referred to as Li's criterion, in terms of the non-negativity of a particular
infinite sequence of real numbers. We formulate the analogue of Li's criterion as
an equivalence for the generalized quasi-Riemann hypothesis for functions in an
extension of the Selberg class, and give arithmetic formulae for the corresponding
Li coefficients in terms of parameters of the function in question. Moreover, we
give explicit non-negative bounds for certain sums of special values of polygamma
functions, involved in the arithmetic formulae for these Li coefficients, for a wide class of functions. Finally, we discuss an existing result on correspondences between
zero-free regions and the non-negativity of the real parts of finitely many Li
coefficients. This discussion involves identifying some errors in the original source work which seem to render one of its theorems conjectural. Under an appropriate
conjecture, we give a generalization of the result in question to the case of Li coefficients corresponding to the generalized quasi-Riemann hypothesis. We also
give a substantial discussion of research on Li's criterion since its inception, and
some additional new supplementary results, in the first chapter.

► The objective of this thesis is to study the distribution of the number of principal ideals generated by an irreducible element in an algebraic…
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▼ The objective of this thesis is to study the distribution of the number of principal ideals generated by an irreducible element in an algebraic number field, namely in the non-unique factorization ring of integers of such a field. In particular we are investigating the size of M(x), defined as M ( x ) =∑ (α) α irred.|N (α)|≤≠ 1, where x is any positive real number and N (α) is the norm of α. We finally obtain asymptotic results for hl(x).
Advisors/Committee Members: William Snyder, Henrik Bresinsky, Ali Ozluk.

Rozario, R. (2003). The Distribution of the Irreducibles in an Algebraic Number Field. (Masters Thesis). University of Maine. Retrieved from https://digitalcommons.library.umaine.edu/etd/405

Chicago Manual of Style (16th Edition):

Rozario, Rebecca. “The Distribution of the Irreducibles in an Algebraic Number Field.” 2003. Masters Thesis, University of Maine. Accessed June 07, 2020.
https://digitalcommons.library.umaine.edu/etd/405.

Rozario R. The Distribution of the Irreducibles in an Algebraic Number Field. [Internet] [Masters thesis]. University of Maine; 2003. [cited 2020 Jun 07].
Available from: https://digitalcommons.library.umaine.edu/etd/405.

Council of Science Editors:

Rozario R. The Distribution of the Irreducibles in an Algebraic Number Field. [Masters Thesis]. University of Maine; 2003. Available from: https://digitalcommons.library.umaine.edu/etd/405

Este trabalho Ã baseado no artigo Finiteness of the class group of a number field via lattice packings. Daremos aqui uma prova alternativa da finitude…
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▼

Este trabalho Ã baseado no artigo Finiteness of the class group of a number field via lattice packings. Daremos aqui uma prova alternativa da finitude do grupo das classes de um corpo de nÃmeros de grau n. Ela Ã baseada apenas no fato de que a densidade de centro de um empacotamento reticulado n-dimensional Ã limitado fora do infinito.

This work is based on the article Finiteness of the class group of a number field via lattice packings. An alternative proof of the finiteness of the class group of a number field of the degree n is presented. It is based solely on the fact that the center density of an n-dimensional lattice packing is bounded away from infinity.

► This master’s thesis, Global field isomorphisms: a class field theoretical approach, was written by Harry Smit from October 2015 until June 2016. It is submitted…
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▼ This master’s thesis, Global field isomorphisms: a class field theoretical approach, was written
by Harry Smit from October 2015 until June 2016. It is submitted to the Department
of Mathematics at Utrecht University. The research was conducted under supervision
of professor Gunther Cornelissen, and the second reader is professor Frits Beukers.
After an introduction into both local and global class field theory, we investigate two
objects that uniquely determine the isomorphism type of a global field K, following
an unpublished article of Cornelissen, Li, and Marcolli. Firstly, we use the maximal
abelian Galois group to create a topological space X_K and subsequently a dynamical
system by defining an action of the integral ideals I_K on X_K. Secondly, we combine
the maximal abelian Galois group with the Dirichlet L-series. Both these objects
can be described using only objects from within K itself. The original contributions in
this thesis are the proof that X_K is a Hausdorff space and various improvements on
the proofs given by Cornelissen, Li, and Marcolli.
Advisors/Committee Members: Cornelissen, G.L.M..

► Quintic abelian fields are characterized in terms of their conductor and a certain Galois group. From these, a generating polynomial and its roots and an…
(more)

▼ Quintic abelian fields are characterized in terms of their
conductor and a certain Galois group.
From these, a generating polynomial and its roots and an
integral basis are computed.
A method for finding the fundamental units, regulators and
class numbers is then developed.
Tables listing the coefficients of a generating polynomial,
the regulator, the classnumber, and a coefficients of a
fundamental unit are given for 1527 quintic abelian fields.
Of the seven cases where the class group structure is not
immediate from the classnumber, six have their structure
computed.
Advisors/Committee Members: Parry, Charles J. (committeechair), Floyd, William J. (committee member), Johnson, Lee W. (committee member), Brown, Ezra A. (committee member), Ball, Joseph A. (committee member).

In this thesis, we focus on class group computations in number fields. We start by describing an algorithm for reducing the size of a defining polynomial of a number field. There exist infinitely many polynomials that define a specific number field, with arbitrarily large coefficients, but our algorithm constructs the one that has the absolutely smallest coefficients. The advantage of knowing such a ``small'' defining polynomial is that it makes calculations in the number field easier because smaller values are involved. In addition, thanks to such a small polynomial, one can use specific algorithms that are more efficient than the general ones for class group computations. The generic algorithm to determine the structure of a class group is based on ideal reduction, where ideals are viewed as lattices. We describe and simplify the algorithm presented by Biasse and Fieker in 2014 at ANTS and provide a more thorough complexity analysis for~it. We also examine carefully the case of number fields defined by a polynomial with small coefficients. We describe an algorithm similar to the Number Field Sieve, which, depending on the field parameters, may reach the hope for complexity L(1/3). Finally, our results can be adapted to solve an associated problem: the Principal Ideal Problem. Given any basis of a principal ideal (generated by a unique element), we are able to find such a generator. As this problem, known to be hard, is…

► In this thesis we look at three counting problems connected to orders in number fields. First we study the probability that for a random polynomial…
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▼ In this thesis we look at three counting problems connected to orders in number
fields. First we study the probability that for a random polynomial f in Z[X]
the ring Z[X]/f is the maximal order in Q[X]/f. Connected to this is the
probability that a random polynomial has a squarefree discriminant.
The second counting problem counts the number of subrings within maximal orders.
We know that the number of subrings of given index is finite. We determine
bounds for the number of suborders in terms of the rank of the maximal order and
the index of the suborder. Connected to this is a question from Manjul Bhargava
on the number of suborders in quintic rings.
The final problem deals with class groups. There are bounds known for the classnumber of a maximal order, and we use these bounds to bound the classnumber of
general orders.

► Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of…
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▼ Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number λ(f). The number λ(f) is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur.
Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on Sg with orientable foliations whose stretch factor λ has algebraic degree 2g. Moreover, the stretch factor λ is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d≤g. Our examples also give a new approach to a conjecture of Penner.
Advisors/Committee Members: Margalit, Dan (advisor), Etnyre, John (committee member), Garoufalidis, Stavros (committee member), Ulmer, Douglas (committee member), Wortman, Kevin (committee member).

► If we adjoin the cube root of a cube free rational integer <i>m</i> to the rational numbers we construct a cubic field. If we adjoin…
(more)

▼ If we adjoin the cube root of a cube free rational integer <i>m</i> to the rational numbers we construct a cubic field. If we adjoin the cube roots of distinct cube free rational integers <i>m</i> and <i>n</i> to the rational numbers we construct a bicubic field. The number theoretic invariants for the cubic fields and their normal closures are well known. Some work has been done on the units, classnumbers and other invariants of the bicubic fields and their normal closures by Parry but no method is available for calculating those invariants. This dissertation provides an algorithm for calculating the number theoretic invariants of the bicubic fields and their normal closure. Among these invariants are the discriminant, an integral basis, a set of fundamental units, the classnumber and the rank of the 3-class group.
Advisors/Committee Members: Parry, Charles J. (committeechair), Renardy, Michael J. (committee member), Green, Edward L. (committee member), Linnell, Peter A. (committee member), Brown, Ezra A. (committee member).

Chalmeta, A. P. (2006). On the Units and the Structure of the 3-Sylow Subgroups of the Ideal Class Groups of Pure Bicubic Fields and their Normal Closures. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/29191

Chicago Manual of Style (16th Edition):

Chalmeta, A Pablo. “On the Units and the Structure of the 3-Sylow Subgroups of the Ideal Class Groups of Pure Bicubic Fields and their Normal Closures.” 2006. Doctoral Dissertation, Virginia Tech. Accessed June 07, 2020.
http://hdl.handle.net/10919/29191.

MLA Handbook (7th Edition):

Chalmeta, A Pablo. “On the Units and the Structure of the 3-Sylow Subgroups of the Ideal Class Groups of Pure Bicubic Fields and their Normal Closures.” 2006. Web. 07 Jun 2020.

Vancouver:

Chalmeta AP. On the Units and the Structure of the 3-Sylow Subgroups of the Ideal Class Groups of Pure Bicubic Fields and their Normal Closures. [Internet] [Doctoral dissertation]. Virginia Tech; 2006. [cited 2020 Jun 07].
Available from: http://hdl.handle.net/10919/29191.

Council of Science Editors:

Chalmeta AP. On the Units and the Structure of the 3-Sylow Subgroups of the Ideal Class Groups of Pure Bicubic Fields and their Normal Closures. [Doctoral Dissertation]. Virginia Tech; 2006. Available from: http://hdl.handle.net/10919/29191

25.
Ward, Kenneth A.Asymptotics Of ClassNumber And Genus For Abelian Extensions Of An Algebraic Function Field.

…Fg
F
n
In general, the asymptotic relationship between classnumber and genus remains
an… …for finite normal extensions K of Q with classnumber hK , regulator RK , and
discriminant… …The work of Inaba depends upon an estimate for the number of integral divisors of
degree… …extension is equal to Fq [38].
By use of a result in class field theory that employs the… …global class field theory that
the Artin map of a finite, unramified, and abelian extension of…

► The main class="hilite">subject of this thesis is the CM classnumber one problem for curves of genus g, in the cases g=2 and g=3. The…
(more)

▼ The main class="hilite">subject of this thesis is the CM classnumber one problem for curves of genus g, in the cases g=2 and g=3. The problem asks for which CM fields of degree 2g with a primitive CM type are the corresponding CM curves of genus g defined over the reflex field.
Chapter 1 is an introduction to abelian varieties and complex multiplication theory. We present facts that we will use in later chapters. The results in this chapter are mostly due to Shimura and Taniyama.
Chapter 2 is a joint work with Marco Streng, we give a solution to the CM classnumber one problem for curves of genus 2.
Chapter 3 deals with the CM classnumber one problem for curves of genus 3. We give a partial solution to this problem. We restrict ourselves to the case where the sextic CM field corresponding to such a curve contains an imaginary quadratic subfield.
Chapter 4 gives the complete list of sextic CM fields K for which there
exist principally polarized simple abelian threefolds that has CM by the maximal order of K with rational field of moduli.
Advisors/Committee Members: Supervisor: P. Stevenhagen, Streng, A. Enge Co-Supervisor: T.C..

► We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote…
(more)

▼ We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows.
Let C(D) denote the class groups of the quadratic number field of discriminant D.
Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a2+16c4.
Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4.
These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec.
In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion.
Advisors/Committee Members: Supervisor: P. Stevenhagen, Fouvry, E..

In this work, we study the so called \"Class Field Theory\", which give us a simple description of the abelian extension of local and global elds. We also study some applications, like the Kronecker-Weber and Scholz-Reichardt theorems

► We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we…
(more)

▼ We study the Galois module structure of the ideal ray
class group and the group of units of a real abelian number field.
Specifically, we derive explicit annihilators of the ideal ray
class groups in the vein of the classical Stickelberger theorems.
This is made possible by generalizing a theorem of Rubin which in
turn allows us to describe a relationship between the Galois module
structure of certain explicit quotients of units and the Galois
module structure of the ray class group. Along the way, we're
compelled to study the Galois module structure of the p-adic
completion of the units. We derive numerous conditions under which
we may conclude that this module is cyclic some of which allow for
p to divide the order of the Galois group. Under those conditions,
we are able to relate the annihilators of the p-parts of various
explicit quotients of units to annihilators of the p-parts of the
ray class groups in many cases. This is a generalization of a
theorem of Thaine.
Advisors/Committee Members: Sinnott, Warren (Advisor).

…x5B;R− : S−
k ] = hpn where hpn is the relative classnumber of Q(ζpn ).
1… …is the classnumber of k.
Once again, for general abelian number fields, Sinnott [16… …CHAPTER 1
INTRODUCTION
1.1 History
Let k be an abelian number field of conductor m. Let… …For any real x, let {x} denote the
unique real number such that x − {x}… …the ideal class group of k.
Stickelberger’s theorem follows by studying the factorization of…

All, T. J. (2013). On the Galois module structure of the units and ray classes
of a real abelian number field. (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642

Chicago Manual of Style (16th Edition):

All, Timothy James. “On the Galois module structure of the units and ray classes
of a real abelian number field.” 2013. Doctoral Dissertation, The Ohio State University. Accessed June 07, 2020.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642.

All TJ. On the Galois module structure of the units and ray classes
of a real abelian number field. [Internet] [Doctoral dissertation]. The Ohio State University; 2013. [cited 2020 Jun 07].
Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642.

Council of Science Editors:

All TJ. On the Galois module structure of the units and ray classes
of a real abelian number field. [Doctoral Dissertation]. The Ohio State University; 2013. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642

30.
Sinclair, Brian.
Algorithms for enumerating invariants and extensions of
local fields.

► There are many computationally difficult problems in the study of p-adic fields, among them the classification of field extensions and the decomposition of global ideals.…
(more)

▼ There are many computationally difficult problems in
the study of p-adic fields, among them the classification of field
extensions and the decomposition of global ideals. The main goal of
this work is present efficient algorithms, leveraging the Newton
polygons and residual polynomials, to solve many of these problems
faster and more efficiently than present methods. Considering
additional invariants, we extend Krasner's mass formula,
dramatically improve general extension enumeration using the
reduced Eisenstein polynomials of Monge, and provide a detailed
account of algorithms that compute Okutsu invariants, which have
many uses, through the lens of partitioning the set of zeros of
polynomials.; Algorithms, Local class field theory, Number theory,
P-adic fields
Advisors/Committee Members: Sebastian Pauli (advisor).

…Kronecker's work on the factoring of prime ideals in number elds when he
namely the functions… …properties of a
function by expanding functions locally, should be translatable to number theory… …series expansion of a rational numbernumber
r ∈ Q in terms of powers of a prime
p,
1
r… …X
ri pi .
i=N
Hensel called this series the
p-adic
nal number can be expressed
about… …near
r
near
p,
expansion of r.
p-adically
With respect to a prime number
in this way…

Sinclair, B. (2015). Algorithms for enumerating invariants and extensions of
local fields. (Doctoral Dissertation). University of North Carolina – Greensboro. Retrieved from http://libres.uncg.edu/ir/listing.aspx?styp=ti&id=18186

Chicago Manual of Style (16th Edition):

Sinclair, Brian. “Algorithms for enumerating invariants and extensions of
local fields.” 2015. Doctoral Dissertation, University of North Carolina – Greensboro. Accessed June 07, 2020.
http://libres.uncg.edu/ir/listing.aspx?styp=ti&id=18186.

Sinclair B. Algorithms for enumerating invariants and extensions of
local fields. [Doctoral Dissertation]. University of North Carolina – Greensboro; 2015. Available from: http://libres.uncg.edu/ir/listing.aspx?styp=ti&id=18186